Is this statement true?
Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ is polynomially growing and of infinite order, then the semidirect product $\mathbb{F}\rtimes_\Phi\mathbb{Z}$ is acylindrically hyperbolic but not relatively hyperbolic.
If there is anything unclear about notation, please tell me. This statement is closely related to Problem 8.2 in A. Minasyan and D. Osin. Acylindrical hyperbolicity of groups acting on trees. math. Annalen, 362:1055–1105, 2015 (arXiv link). This problem asks which mapping tori of injective endomorphisms of finitely generated free groups are acylindrically hyperbolic.